What is Cramer's Rule for 3 Variables? | Simple Explanation for Class 10

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What is Cramer's Rule for Three Variables?

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

Cramer's Rule for Three Variables is a clever method to solve a system of three linear equations with three unknowns (like x, y, and z) using determinants. It helps you find the value of each variable by calculating ratios of determinants, making it a systematic way to solve complex problems.

Simple Example

Quick Example

Imagine you went to a snack shop and bought 3 samosas, 2 jalebis, and 1 chai, paying Rs. 70. Your friend bought 2 samosas, 1 jalebi, and 2 chais for Rs. 60. Another friend bought 1 samosa, 3 jalebis, and 1 chai for Rs. 80. Cramer's Rule can help you figure out the exact price of one samosa, one jalebi, and one chai individually.

Worked Example

Step-by-Step

Let's solve the system:
2x + y - z = 8
x - 2y + 3z = -11
3x + 2y + z = 1

---1. Calculate the determinant of the coefficient matrix (D):
D = | 2 1 -1 |
| 1 -2 3 |
| 3 2 1 |
D = 2((-2)(1) - (3)(2)) - 1((1)(1) - (3)(3)) + (-1)((1)(2) - (-2)(3))
D = 2(-2 - 6) - 1(1 - 9) - 1(2 + 6)
D = 2(-8) - 1(-8) - 1(8)
D = -16 + 8 - 8 = -16

---2. Calculate Dx (replace x-coefficients with constants):
Dx = | 8 1 -1 |
| -11 -2 3 |
| 1 2 1 |
Dx = 8((-2)(1) - (3)(2)) - 1((-11)(1) - (3)(1)) + (-1)((-11)(2) - (-2)(1))
Dx = 8(-2 - 6) - 1(-11 - 3) - 1(-22 + 2)
Dx = 8(-8) - 1(-14) - 1(-20)
Dx = -64 + 14 + 20 = -30

---3. Calculate Dy (replace y-coefficients with constants):
Dy = | 2 8 -1 |
| 1 -11 3 |
| 3 1 1 |
Dy = 2((-11)(1) - (3)(1)) - 8((1)(1) - (3)(3)) + (-1)((1)(1) - (-11)(3))
Dy = 2(-11 - 3) - 8(1 - 9) - 1(1 + 33)
Dy = 2(-14) - 8(-8) - 1(34)
Dy = -28 + 64 - 34 = 2

---4. Calculate Dz (replace z-coefficients with constants):
Dz = | 2 1 8 |
| 1 -2 -11 |
| 3 2 1 |
Dz = 2((-2)(1) - (-11)(2)) - 1((1)(1) - (-11)(3)) + 8((1)(2) - (-2)(3))
Dz = 2(-2 + 22) - 1(1 + 33) + 8(2 + 6)
Dz = 2(20) - 1(34) + 8(8)
Dz = 40 - 34 + 64 = 70

---5. Find x, y, and z:
x = Dx / D = -30 / -16 = 15/8
y = Dy / D = 2 / -16 = -1/8
z = Dz / D = 70 / -16 = -35/8

Answer: x = 15/8, y = -1/8, z = -35/8

Why It Matters

Cramer's Rule is crucial for solving systems of equations that appear in many fields. Engineers use it to design bridges and circuits, while scientists in AI/ML apply similar logic to train complex models. Understanding this helps you think systematically, a skill vital for careers in technology, medicine, and research.

Common Mistakes

MISTAKE: Forgetting to change the sign for the middle term when calculating a 3x3 determinant (e.g., a(ei-fh) - b(di-fg) + c(dh-eg)). | CORRECTION: Always remember the alternating signs (+ - +) when expanding a 3x3 determinant along the first row.

MISTAKE: Swapping columns incorrectly when finding Dx, Dy, or Dz (e.g., replacing the x-column with constants for Dy). | CORRECTION: For Dx, replace ONLY the x-coefficient column with the constant terms. Do the same for y and z columns when finding Dy and Dz respectively.

MISTAKE: Making calculation errors, especially with negative numbers, during determinant expansion. | CORRECTION: Double-check each multiplication and subtraction step, especially when dealing with negative values. Use parentheses to keep track of signs.

Practice Questions

Try It Yourself

QUESTION: Solve for x, y, and z using Cramer's Rule:
x + y + z = 6
2y + 5z = -4
2x + 5y - z = 27
| ANSWER: x = 5, y = 3, z = -2

QUESTION: Find the value of 'y' only using Cramer's Rule for the system:
3x - 2y + z = 10
x + 3y - 2z = -5
2x - y + 3z = 13
| ANSWER: y = -1

QUESTION: A local store sells three types of ladoos: Motichoor, Besan, and Boondi. If 2 Motichoor, 1 Besan, and 3 Boondi cost Rs. 130; 1 Motichoor, 2 Besan, and 1 Boondi cost Rs. 90; and 3 Motichoor, 1 Besan, and 2 Boondi cost Rs. 150. Use Cramer's Rule to find the price of one Motichoor ladoo.
| ANSWER: Price of one Motichoor ladoo = Rs. 40

MCQ

Quick Quiz

Which of the following conditions must be true for Cramer's Rule to be applicable to a system of three linear equations?

The determinant of the coefficient matrix (D) must be zero.

The system must have exactly two variables.

The determinant of the coefficient matrix (D) must NOT be zero.

The number of equations must be different from the number of variables.

The Correct Answer Is:

Cramer's Rule works only if the determinant of the coefficient matrix (D) is non-zero. If D is zero, the system either has no unique solution or infinitely many solutions, and Cramer's Rule cannot be used to find a unique solution.

Real World Connection

In the Real World

In India, companies like ISRO use complex systems of equations to track satellites or calculate rocket trajectories. Cramer's Rule, or the principles behind it, helps solve these equations to pinpoint exact locations and movements in space. Similarly, in medical imaging like MRI scans, mathematical methods derived from these principles help reconstruct 3D images of the human body from raw data.

Key Vocabulary

Key Terms

DETERMINANT: A special number calculated from a square matrix. It tells us important properties of the matrix. | COEFFICIENT MATRIX: A matrix formed by the coefficients (numbers in front of x, y, z) of the variables in a system of equations. | SYSTEM OF LINEAR EQUATIONS: A set of two or more linear equations with the same variables. | UNIQUE SOLUTION: When a system of equations has exactly one set of values for the variables that satisfies all equations.

What's Next

What to Learn Next

Great job understanding Cramer's Rule! Next, you can explore 'Matrix Inversion Method for Solving Systems of Equations'. It's another powerful way to solve similar problems, especially when dealing with even larger systems, and will deepen your understanding of linear algebra.